2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 99.9%

Time: 9.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{3}}{2}\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\\ t_3 := \cos t\_2\\ t_4 := \left(0.25 - t\_1\right) \cdot t\_3\\ t_5 := \sin t\_2 \cdot \left(\left(t\_0 \cdot 0.5\right) \cdot -2\right)\\ 2 \cdot \frac{{t\_4}^{3} + {t\_5}^{3}}{{t\_4}^{2} + \left(t\_5 \cdot t\_5 + t\_5 \cdot \left(t\_3 \cdot \left(t\_1 - 0.25\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (/ (sqrt 3.0) 2.0))
        (t_1 (* t_0 t_0))
        (t_2 (* (acos (- (/ g h))) 0.3333333333333333))
        (t_3 (cos t_2))
        (t_4 (* (- 0.25 t_1) t_3))
        (t_5 (* (sin t_2) (* (* t_0 0.5) -2.0))))
   (*
    2.0
    (/
     (+ (pow t_4 3.0) (pow t_5 3.0))
     (+ (pow t_4 2.0) (+ (* t_5 t_5) (* t_5 (* t_3 (- t_1 0.25)))))))))

C

double code(double g, double h) {
	double t_0 = sqrt(3.0) / 2.0;
	double t_1 = t_0 * t_0;
	double t_2 = acos(-(g / h)) * 0.3333333333333333;
	double t_3 = cos(t_2);
	double t_4 = (0.25 - t_1) * t_3;
	double t_5 = sin(t_2) * ((t_0 * 0.5) * -2.0);
	return 2.0 * ((pow(t_4, 3.0) + pow(t_5, 3.0)) / (pow(t_4, 2.0) + ((t_5 * t_5) + (t_5 * (t_3 * (t_1 - 0.25))))));
}

Fortran

real(8) function code(g, h)
    real(8), intent (in) :: g
    real(8), intent (in) :: h
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    t_0 = sqrt(3.0d0) / 2.0d0
    t_1 = t_0 * t_0
    t_2 = acos(-(g / h)) * 0.3333333333333333d0
    t_3 = cos(t_2)
    t_4 = (0.25d0 - t_1) * t_3
    t_5 = sin(t_2) * ((t_0 * 0.5d0) * (-2.0d0))
    code = 2.0d0 * (((t_4 ** 3.0d0) + (t_5 ** 3.0d0)) / ((t_4 ** 2.0d0) + ((t_5 * t_5) + (t_5 * (t_3 * (t_1 - 0.25d0))))))
end function

Java

public static double code(double g, double h) {
	double t_0 = Math.sqrt(3.0) / 2.0;
	double t_1 = t_0 * t_0;
	double t_2 = Math.acos(-(g / h)) * 0.3333333333333333;
	double t_3 = Math.cos(t_2);
	double t_4 = (0.25 - t_1) * t_3;
	double t_5 = Math.sin(t_2) * ((t_0 * 0.5) * -2.0);
	return 2.0 * ((Math.pow(t_4, 3.0) + Math.pow(t_5, 3.0)) / (Math.pow(t_4, 2.0) + ((t_5 * t_5) + (t_5 * (t_3 * (t_1 - 0.25))))));
}

Python

def code(g, h):
	t_0 = math.sqrt(3.0) / 2.0
	t_1 = t_0 * t_0
	t_2 = math.acos(-(g / h)) * 0.3333333333333333
	t_3 = math.cos(t_2)
	t_4 = (0.25 - t_1) * t_3
	t_5 = math.sin(t_2) * ((t_0 * 0.5) * -2.0)
	return 2.0 * ((math.pow(t_4, 3.0) + math.pow(t_5, 3.0)) / (math.pow(t_4, 2.0) + ((t_5 * t_5) + (t_5 * (t_3 * (t_1 - 0.25))))))

Julia

function code(g, h)
	t_0 = Float64(sqrt(3.0) / 2.0)
	t_1 = Float64(t_0 * t_0)
	t_2 = Float64(acos(Float64(-Float64(g / h))) * 0.3333333333333333)
	t_3 = cos(t_2)
	t_4 = Float64(Float64(0.25 - t_1) * t_3)
	t_5 = Float64(sin(t_2) * Float64(Float64(t_0 * 0.5) * -2.0))
	return Float64(2.0 * Float64(Float64((t_4 ^ 3.0) + (t_5 ^ 3.0)) / Float64((t_4 ^ 2.0) + Float64(Float64(t_5 * t_5) + Float64(t_5 * Float64(t_3 * Float64(t_1 - 0.25)))))))
end

MATLAB

function tmp = code(g, h)
	t_0 = sqrt(3.0) / 2.0;
	t_1 = t_0 * t_0;
	t_2 = acos(-(g / h)) * 0.3333333333333333;
	t_3 = cos(t_2);
	t_4 = (0.25 - t_1) * t_3;
	t_5 = sin(t_2) * ((t_0 * 0.5) * -2.0);
	tmp = 2.0 * (((t_4 ^ 3.0) + (t_5 ^ 3.0)) / ((t_4 ^ 2.0) + ((t_5 * t_5) + (t_5 * (t_3 * (t_1 - 0.25))))));
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(N[Sqrt[3.0], $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.25 - t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[t$95$2], $MachinePrecision] * N[(N[(t$95$0 * 0.5), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(N[Power[t$95$4, 3.0], $MachinePrecision] + N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$4, 2.0], $MachinePrecision] + N[(N[(t$95$5 * t$95$5), $MachinePrecision] + N[(t$95$5 * N[(t$95$3 * N[(t$95$1 - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 98.4%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} + {\left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right)}^{3}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(\left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right) - \left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right)\right)}} \]

5. Final simplification 100.0%

   \[\leadsto 2 \cdot \frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)}^{3} + {\left(\sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot -2\right)\right)}^{3}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(\left(\sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot -2\right)\right) \cdot \left(\sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot -2\right)\right) + \left(\sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right) \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot -2\right)\right) \cdot \left(\cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right) \cdot \left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} - 0.25\right)\right)\right)} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\\ -\mathsf{fma}\left(\sqrt{3}, \sin t\_0, \cos t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* (acos (- (/ g h))) 0.3333333333333333)))
   (- (fma (sqrt 3.0) (sin t_0) (cos t_0)))))

C

double code(double g, double h) {
	double t_0 = acos(-(g / h)) * 0.3333333333333333;
	return -fma(sqrt(3.0), sin(t_0), cos(t_0));
}

Julia

function code(g, h)
	t_0 = Float64(acos(Float64(-Float64(g / h))) * 0.3333333333333333)
	return Float64(-fma(sqrt(3.0), sin(t_0), cos(t_0)))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, (-N[(N[Sqrt[3.0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision] + N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} + {\left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right)}^{3}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(\left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right) - \left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right) \cdot \left(-2 \cdot \left(\frac{\sqrt{3}}{2} \cdot 0.5\right)\right)\right)\right)}} \]

6. Applied rewrites 100.0%

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(\sqrt{3} \cdot -0.5, \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), -0.5 \cdot \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)} \]

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) + \frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right)} \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot 0.3333333333333333\right), \cos \left(\cos^{-1} \left(\frac{-g}{h}\right) \cdot 0.3333333333333333\right)\right)} \]

10. Final simplification 100.0%

    \[\leadsto -\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))